Approximation Metrics for Discrete and Continuous Systems

1. VERIMAG Approximation Metrics forDiscrete and Continuous Systems Antoine Girard and George J. Pappas Antoine.Girard@imag.fr, pappasg@ee.upenn.edu Workshop “Topics in Computation and Control”March 27th 2006, Santa Barbara, CA, USA

2. Safety Verification Safe Unsafe A general system S with observations: • Language of S: set of observed trajectories of S. • Reachable set of S: subset of observations reached by trajectories of S. • Safety verification problem or Reachability problem

3. What is Abstraction? S2 is safe  S1 is safe Given a (complicated) system S1, we compute a (simple) system S2: All the trajectories of S1 are trajectories of S2. (i.e. L(S1)  L(S2)). Then, Reach(S1)  Reach(S2).

4. Hierarchy of Abstraction Bisimulation relation: S1 S2 Simulation relation: S1 S2 Language equivalence: L(S1) = L(S2) Language inclusion: L(S1)  L(S2) Reachability equivalence: Reach(S1) = Reach(S2) Reachability inclusion: Reach(S1)  Reach(S2)

5. From Abstraction to Approximation • The previous notions of abstraction are all exact: • When dealing with continuous and hybrid systems: • - Uncertain parameters, • - Noisy inputs. • Notions of abstraction become restrictive and not robust. • Notions of approximation seem more appropriate. • Notions of approximation need metrics. Each trajectory of S1 is a trajectory of S2. Each trajectory of S1 has a neighboring trajectory of S2.

6. Outline of the Talk • Approximation metrics for transition systems • - Hierarchy of approximation metrics • - Computational framework • 2. Applications to safety verification: • - Approximation of continuous systems • - Safety verification using simulation

7. Transition Systems • A transition system consists of A set of states Q A subset of initial states Q0 Q A set of events Σ The transition relation A set of observations Π The observation mapq = π • We assume systems to be non-blocking, possibly nondeterministic. The sets Q, Σ, and Πmay be infinite. • Modeling framework for discrete, continuous and hybrid systems.

8. A quantitative theory of approximations requires metrics. A transition system is a called metric transition system if The set of states has a metric dQ : Q x Q  R+ The set of events has the discrete metric The set of observations has a metric dΠ : Q x Q  R+ + some regularity assumptions. Metric Transition Systems

9. Relevant question for the safety verification problem: Since Reach(S1), Reach(S2)  Π which is a metric space where h, h denote Hausdorff distances. Reachability Metrics How well Reach(S1) is approximated by Reach(S2) ?

10. Application to Safety Verification Any S2, such that dR(S1,S2)  η/2, allows to verify that S1 is safe where η = dist(Reach(S1),ΠU). The more robustly safe S1, the more approximations are allowed, the easier the safety verification. Reach(S1)  N(Reach(S2),δ) where δ = dR(S1,S2) Reach(S2)  N(ΠU,δ) =   Reach(S1)  ΠU = 

11. More complex properties: language approximation is more appropriate. Lifting the metric dΠ to sequences (in the infinity sense): Reachability and language metrics are useful but difficult to compute. Language Metrics

12. Approximate Simulation • Consider two transition systems and let δ 0 be given • R  Q1 x Q2 is a δ - approximate simulation relation if it 1. respects observations: if (q1,q2)  R then dΠ(q11, q22)  δ 2. respects transitions: if (q1,q2)  R then • For δ = 0, we recover the usual notion of exact simulation.

13. If  q1  Q10,  q2  Q20 such that (q1,q2)  R then we say that Tightest precision with which S2 approximately simulates S1 Simulation metric Under some regularity assumptions: Simulation Metric S2 approximately simulates S1 with precision δ: S1 δS2

14. Symmetric version of approximate simulation: approximate bisimulation Tightest precision with which S1 and S2 are approximately bisimilar  Bisimulation metric Under some regularity assumptions: Bisimulation Metric

15. Hierarchy of Approximation Metrics Bisimulation metric: dB(S1,S2) Simulation metric: dS(S1,S2) Undirected language metric: dL(S1,S2) Directed language metric: dL(S1,S2) Undirected reachability metric: dR(S1,S2) Directed reachability metric: dR(S1,S2) A. Girard, G.J. Pappas, Approximation metrics for discrete and continuous systems, TAC, accepted.

16. Zero Sections Bisimulation relation: S1 S2 Simulation relation: S1 S2 Language equivalence: cl(L(S1)) = cl(L(S2)) Language inclusion: cl(L(S1))  cl(L(S2)) Reachability equivalence: cl(Reach(S1)) = cl(Reach(S2)) Reachability inclusion: cl(Reach(S1))  cl(Reach(S2))

17. How do we compute of the simulation and bisimulation metrics ? Dual approach to the relations based on functions: A (bi)-simulation function is a function V: Q1 x Q2  R+  { + }, RV(δ) = { (q1,q2) | V (q1,q2)  δ } is a δ-approximate (bi)-simulation relation Then, the (bi)-simulation metrics can be bounded by Computational Framework

18. Characterization of bisimulation functions: Minimal bisimulation function: smallest function satisfying equation For the minimal bisimulation function Minimal bisimulation function hard to compute for infinite state systems. Bisimulation functions

19. Outline of the Talk • Approximation metrics for transition systems • - Hierarchy of approximation metrics • - Computational framework • 2. Applications to safety verification: • - Approximation of continuous systems • - Safety verification using simulation

20. Continuous Dynamics S generates the transition system T = (Q, Q0, Σ, , Π, . ) where The set of states Q = Rn The subset of initial states Q0 = I The set of labels Σ = R+ The transition relation is given by The set of observations Π = Rp The observation mapx = g(x)

21. Bisimulation functions is a bisimulation function if and only if

22. Example Bisimulation function:

23. Example Indeed, And Then, Since ,

24. Constrained Linear Systems For bisimulation functions of the form we get

25. We search bisimulation functions of the form Decomposition transient/asymptotic error Characterization Truncated Quadratic Functions For some λ > 0. A. Girard, G.J. Pappas, Approximate bisimulations for constrained linear systems, CDC 2005.

26. Truncated Quadratic Functions • Universal for stable constrained linear systems: • Two stable constrained linear systems are approximately bisimilar.(but the precision can be very bad!) • Characterization allows to derive computationally effective algorithms. • Generalizable to non-stable systems: • two systems are approximately bisimilar ifftheir unstable subsystems are exactly bisimilar.

27. MATISSE Metrics for Approximate TransItion Systems Simulation and Equivalence • MATLAB toolbox • Functionalities: • - Computes a bisimulation function between a system and its projection. • - Evaluates the bisimulation distance between a system and its projection. • - Finds a good projection of a system (given the desired dimension). • - Performs reachability computations using zonotopes. • Available at • http://www.seas.upenn.edu/~agirard/Software/MATISSE/index.html

28. MATISSE Metrics for Approximate TransItion Systems Simulation and Equivalence Example of application: safety verification of a ten-dimensional system 10-dimensionaloriginal system 5-dimensionalapproximation 7-dimensionalapproximation A. Girard, G.J. Pappas, Approximate bisimulation relations for constrained linear systems, Submitted 2005.

29. Extensions • Computational method for nonlinear autonomous systems (SOS) • Characterization of approximate simulation for hybrid systems • Theoretical framework, computational methods for stochastic linear dynamical/hybrid systems (with stochastic jumps) A. Girard, G.J. Pappas, Approximate bisimulations for nonlinear dynamical systems, CDC 2005. A. Girard, A.A Julius, G.J. Pappas, Approximate simulation relations for hybrid systems, ADHS 2006. A.A. Julius, A. Girard, G.J. Pappas, Approximate bisimulation for a class of stochastic hybrid systems, ACC 2006. Talk on Wednesday: A.A. Julius, Approximate abstraction of stochastic hybrid automata, HSCC 2006.

30. Outline of the Talk • Approximation metrics for transition systems • - Hierarchy of approximation metrics • - Computational framework • 2. Applications to safety verification: • - Approximation of continuous systems • - Safety verification using simulation

31. Let us consider a metric transition system A pseudo-metric dB on the set of states Q: dB(q, q) = 0 dB(q1, q2) = dB(q2,q1) dB(q1, q3)  dB(q1, q2) + dB(q2, q3) is a bisimulation metric if there exists  > 1 Bisimulation metric  pseudo-metric + bisimulation function. Back to Transition Systems

32. Simulation-based Reachability • The bisimulation metric allows to sample subsets of Q • Simulation-based reachability : • - sample the set of initial states • - sample of the successor operators

33. Simulation-based Reachability q01 q02 Q0 Post(q01) Post(q02) q11 q12 q13 • Simulation-based reachability: let δδ/λ + ε

34. Simulation-based Reachability q01 q02 Q0 q11 q12 q13 • Because d is a bisimulation metric we can show that • Then, it follows that Talk on Friday: A. Girard, G.J. Pappas, Verification using Simulation, HSCC 2006.

35. Conclusion • Unified (discrete/continuous) framework for system approximation. • Approximation as a relaxation of abstraction:- metrics instead of relations.-more significant complexity reduction. • Approach based on bisimulation functions- Lyapunov like characterization- computational methods (LMIs, SOS, Games) • Robustness of the safety of the original system is critical for the amount of approximations that can be done.